Fungrim entry: 3fe68f

Symbol: OpenInterval —

\left(a, b\right)

— Open interval

►OpenInterval(a, b) —

\left(a, b\right)

— Represents

\left\{ x : x \in \mathbb{R} \cup \left\{-\infty, \infty\right\} \,\mathbin{\operatorname{and}}\, a < x < b \right\}

►OpenInterval(0, 1) —

\left(0, 1\right)

— Represents the open unit interval.

►OpenInterval(1, 1) —

\left(1, 1\right)

— Represents the empty set.

►OpenInterval(Neg(Infinity), 0) —

\left(-\infty, 0\right)

— Represents half the extended real line (excluding minus infinity and zero).

►OpenInterval(1, -1) —

\left(1, -1\right)

— Represents the empty set. Note: potentially confusing rendering.

►Add(1, Mul(OpenInterval(0, 1), ConstI)) —

1 + \left(0, 1\right) i

— Represents a set of points in the complex plane. OpenInterval(a, b) should only be used with extended real number

a

and

b

as endpoints, but line segments in the complex plane can be constructed by applying arithmetic operations to a set of real numbers (acting pointwise).

►Add(OpenInterval(1, 4), Mul(OpenInterval(0, 1), ConstI)) —

\left(1, 4\right) + \left(0, 1\right) i

— Represents a rectangle in the complex plane.

Definitions:

Fungrim symbol	Notation	Short description
`OpenInterval`	$\left(a, b\right)$	Open interval
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("3fe68f"),
    SymbolDefinition(OpenInterval, OpenInterval(a, b), "Open interval"),
    CodeExample(OpenInterval(a, b), "Represents", Set(x, ForElement(x, Union(RR, Set(Neg(Infinity), Infinity))), Less(a, x, b)), "."),
    CodeExample(OpenInterval(0, 1), "Represents the open unit interval."),
    CodeExample(OpenInterval(1, 1), "Represents the empty set."),
    CodeExample(OpenInterval(Neg(Infinity), 0), "Represents half the extended real line (excluding minus infinity and zero)."),
    CodeExample(OpenInterval(1, -1), "Represents the empty set.", " Note: potentially confusing rendering."),
    CodeExample(Add(1, Mul(OpenInterval(0, 1), ConstI)), "Represents a set of points in the complex plane. ", SourceForm(OpenInterval(a, b)), "should only be used with extended real number", a, "and", b, "as endpoints, but line segments in the complex plane can be constructed by applying arithmetic operations to a set of real numbers (acting pointwise)."),
    CodeExample(Add(OpenInterval(1, 4), Mul(OpenInterval(0, 1), ConstI)), "Represents a rectangle in the complex plane. "))

Topics using this entry

Numbers and infinities